Solving linear recurrence equations

computing: from decision algorithms to constraint programming with multisets, sets, and maps", "Constraint Based Verification of Reactive Systems", and "Zeta and L Functions and Diophantine Problems in Number Theory." i

IEEE Transactions on Acoustics, Speech, and Signal Processing

Optimal linear-time algorithms for solving recurrence equations on simple systolic arrays are presented. The systolic arrays use only one-way communication between processors and communicate with the external environment through only one 1 / 0 port. Because of their architectural simplicity, the arrays are well suited for direct VLSI implementation. Applications to some pattern recognition and sequence comparison problems are given. For example, it is shown that the set of (k + 2)-tuples of strings (x1,. .. , x k + ,, y) such that y is a shuffle of x],. .. , x p + can be recognized by a one-way k-dimensional systolic array in (k + 1) ' nk time. The longest common subsequence (LCS) problem and the string-to-string correction problem are also considered: the length of an LCS of k + 1 sequences can be computed by a one-way k-dimensional systolic array in (k + 1) nk time; the edit distance between two strings can be computed by a one-way dimensional systolic array in 2n-1 time. Applications to other related problems, e.g., dynamic time warping and optimum generalized alignment, as well as optimal-time simulations of multihead acceptors and multitape transducers are also given.

2010

The goal in this paper is to find closed form solutions for linear recurrence equations, by transforming an input equation L to an equation Ls with known solutions. The main problem is how to find a solved equation Ls to which L can be reduced. We solve this problem by computing local

Partial Evaluation and Semantic-Based Program Manipulation, 2006

Journal of Mathematical Physics, 1996

A general method to map a polynomial recursion on a matrix linear one is suggested. The solution of the recursion is represented as a product of a matrix multiplied by the vector of initial values. This matrix is product of transfer matrices whose elements depend only on the polynomial and not on the initial conditions. The method is valid for systems of polynomial recursions and for polynomial recursions of arbitrary order. The only restriction on these recurrent relations is that the highest-order term can be written in explicit form as a function of the lowerorder terms ͑existence of a normal form͒. A continuous analog of this method is described as well.

Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06, 2006

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer N (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in N . We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree N . We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in O( √ N log 2 N ) bit operations; a deterministic one that computes a compact representation of the solution in O(N log 3 N ) bit operations. Similar speedups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.

Journal of VLSI Signal Processing, 1989

The parallelization of many algorithms can be obtained using space-time transformations which are applied on nested do-loops or on recurrence equations. In this paper, we analyze systems of linear recurrence equations, a generalization of uniform recurrence equations. The first part of the paper describes a method for finding automatically whether such a system can be scheduled by an affine timing function, independent of the size parameter of the algorithm. In the second part, we describe a powerful method that makes it possible to transform linear recurrences into uniform recurrence equations. Both parts rely on results on integral convex polyhedra. Our results are illustrated on the Gauss elimination algorithm and on the Gauss-Jordan diagonalization algorithm.

Applied Mathematics and Computation, 1987

A modified version of Miller's and Olver's algorithms based on the "double sweep" method is presented. It is very stable in both directions and automatically controls the maximum number of steps to be used.

2013

The paper is devoted to the methods of solving simultaneous recurrences. Specifically, we discuss transformation of matrix recurrences to regular recurrences and propose a way of solving special matrix recurrences of order three by their decomposition to matrix recurrences of order two.

Computers & Mathematics with Applications, 1991

We present in this paper a three-phase parallel algorithm on the unshtflRe network for solving linear recurrences. Through a detailed analysis on the special nmtrix multiplications involved in the computation we show that the first n terms of an ruth order linear recurrence can be computed in O(m 3 log~) time -ling O(m---l~) processors. For the usual case when m is a small constant, the algorithm achieves cost optinmlity.